The title is Botchan, which is also the name of the main character. Unusually for a Japanese novel, it has been translated into English several times; my version is the most recent one to be published in book form. The preceding translation, done in the 1960s, is quite good, but the earlier ones, which seem to have been done by people who were not native speakers of English, are not.
Somehow I forgot there is actually a brief mention of bikes in this book! Interestingly, it's a comment about people who ride without brakes -- made a hundred years before the hipster/fixie phenomenon. The author of this novel also wrote a cute satirical sketch about learning to ride a bike as a Japanese newbie dealing with an LBS from hell in London circa 1900 - you can find a translation on Google Books by searching for Natsume and "bicycle diary"

As for some other books, Haruki Murakami has already been suggested and he's a good one. A Wild Sheep Chase and After the Quake are favorites of mine; if you want something more romantic there's Norwegian Wood but I don't think you can go wrong with any of his stuff. By the way, he also rides, but mainly to train for triathlons (about which he has also written a non-fiction book).

You might also want to take a look at The Makioka Sisters by Tanizaki, Kokoro by Natsume (same author as Botchan), or, if you're interested in something more contemporary by a female writer, Banana Yoshimoto's Kitchen or Yoko Tawada's The Bridegroom was a Dog. The latter is pretty weird but rewarding and fun -- and the main character does arrive in town on a mountain bike.

Happy reading!